Question

Show that the rate of change of speed can be expressed as $\dfrac{dv}{dt}=\vec{v}.\dfrac{{\vec{a}}}{v}$

Answer 1

Hint: In the above question, we need to assume a particle in two-dimensional motion moving with some velocity. Taking the velocity vectors in both vertical and horizontal directions, we can solve for the required statement.

Complete step by step answer:
Let us consider a particle in two-dimensional motion moving with velocity v.
The vertical and horizontal components of velocity will be respectively ${{v}_{x}},{{v}_{y}}$
The velocity of the body can be written as,
$v=\sqrt{{{v}_{x}}^{2}+{{v}_{y}}^{2}}$
Differentiating the baize equation, we get,
$\begin{align}
  & v=\sqrt{{{v}_{x}}^{2}+{{v}_{y}}^{2}} \\
 & \Rightarrow \dfrac{dv}{dt}=\dfrac{1}{2\sqrt{{{v}_{x}}^{2}+{{v}_{y}}^{2}}}\times (\dfrac{d}{dt}{{v}_{x}}^{2}+\dfrac{d}{dt}{{v}_{y}}^{2}) \\
 & \Rightarrow \dfrac{dv}{dt}=\dfrac{{{v}_{x}}{{a}_{x}}+{{v}_{y}}{{a}_{y}}}{\sqrt{{{v}_{x}}^{2}+{{v}_{y}}^{2}}}=\vec{v}\dfrac{{\vec{a}}}{v} \\
\end{align}$
In this way, we can solve the equation.

Additional information:
Compared to displacement and velocity, acceleration is totally different. It can be violent; some people are scared of it; and if it's big, it forces you to take notice. That feeling you get when you're sitting in a plane during take-off, or slamming on the brakes in a car, or turning a corner at a high speed in a go kart are all situations where you are accelerating. Acceleration is the name we give to any process where the velocity changes. Since velocity is a speed and a direction, there are only two ways for you to accelerate: change your speed or change your direction—or change both. Sometimes an accelerating object will change its velocity by the same amount each second. This is referred to as a constant acceleration since the velocity is changing by a constant amount each second. An object with a constant acceleration should not be confused with an object with a constant velocity and an object with a constant velocity is not accelerating.

Note:
If an object is changing its velocity whether by a constant amount or varying amount, then it is called an acceleration object. An object with constant velocity is not accelerating at all. Velocity is the rate of change of position whereas acceleration is the rate of change of velocity.